The \(D\) –dimensional integral over an Euclidean momentum \(k_E\)  is given by, cf. e.g. [1]},
\(\int \frac{d^D \vec{k}_E}{(2\pi )^D} \frac{(\vec{k}^2_E)^r}{(\vec{k}^2_E + R^2)^m} = \frac{(-1)^{r-m}}{(4\pi )^{D/2}}\frac{\Gamma (r+D/2)}{\Gamma (D/2)} \Gamma (m-r-D/2) (R^2)^{D/2+r-m}.\) 
